Very confusing to figure out with out seeing the actual files.
I allowed it to download to see what I get. From what I can tell the file is a standard xhtml file but why it is named with the extention of .part I do not know.
Take a look at your directory and see if you can find that file listed. Then download the index.html file and view it with a text editor. See if the code is the same as you wrote.
Next thing is to look for a htaccess file using a ftp editor that can see hidden files. Unless you specifically need the htaccess file go ahead and delete it from the directory.

The thing is I’ve always had it working fine for months, and this only happens in the last couple of days.

My best guess is they recently setup some new setting on the server and it mess up this domain… There was also a bug about a week ago that affected all of my PHP sites, I was able to fix it myself but this time it’s really out of my reach.